Algorithms for the Minimal Rational Fraction Representation of Sequences Revisited

Given a binary sequence with length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, determining its minimal rational fraction representation (MRFR) has important applications in the design and cryptanalysis of stream ciphers. There are ma...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 68; no. 2; pp. 1316 - 1328
Main Authors: Che, Jun, Tian, Chengliang, Jiang, Yupeng, Xu, Guangwu
Format: Journal Article
Language:English
Published: New York IEEE 01.02.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Adaptive algorithms
Algorithms
Approximation algorithms
Ciphers
Complexity
Complexity theory
Cryptography
Encryption
extended euclidean algorithm
lattice
Lattices
minimal basis
rational fraction representation
Representations
Sequence
Terminology
Time complexity
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:Given a binary sequence with length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, determining its minimal rational fraction representation (MRFR) has important applications in the design and cryptanalysis of stream ciphers. There are many studies of this problem since Klapper and Goresky first introduced an adaptive rational approximation algorithm with a time complexity of <inline-formula> <tex-math notation="LaTeX">O(n^{2}\log n\log \log n) </tex-math></inline-formula>. In this paper, we revisit this problem by considering both adaptive and non-adaptive efficient algorithms. Compared with the state-of-art methods, we make several contributions to the problem of finding MRFR. Firstly, we find a general and precise recursive relationship between the minimal bases for two adjacent lattices generated by successive truncation sequences. This enables us to improve the currently fastest adaptive algorithm proposed by Li et al. . Secondly, by optimizing a time-consuming step of the well-known Lagrange reduction algorithm for 2-dimensional lattices, we obtain a non-adaptive, and yet practically faster MRFR-solving algorithm named global Euclidean algorithm. Thirdly, we identify theoretical flaws on some non-adaptive methods in the literature by counter-examples and correct the problems by designing modified Euclidean algorithm named partial Euclidean algorithm. Meanwhile, we further reduce the time complexity of existing algorithm from <inline-formula> <tex-math notation="LaTeX">O(n^{2}) </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">O(n\log ^{2}n\log \log n) </tex-math></inline-formula> by invoking the half-gcd algorithm. We also conduct a comprehensive experimental comparative analysis on the above algorithms to validate our theoretical analysis.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3125988